Subgroups — C₃×D₅×#494 #14438794

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×Q₁₆	C₂	1	4	yes	maximal
2	C₃×D₅×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65470	C₂	1	4	yes	maximal, complemented
5	C₃×#65381	C₂	1	2	yes	maximal, complemented
6	C₃×#64988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
8	D₅×#494	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×Q₈×D₅	C₂²	1	4	yes	—
10	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃×#4090	C₂²	1	4	yes	—
13	C₂×C₃×#689	C₂²	1	8	yes	—
14	C₃×D₅×Q₁₆	—	2	16	no	—
15	C₂×C₃×Dic₂₀	C₂²	1	4	yes	—
16	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
17	C₃×D₅×C₄⋊C₄	—	2	4	no	—
18	C₃×#4034	—	2	4	no	—
19	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
20	C₃×#4015	C₂²	1	2	yes	—
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×C₈×Dic₅	C₂²	1	2	yes	—
23	C₃×C₄×#168	C₂²	1	1	yes	characteristic
24	C₂×C₃×#494	—	5	1	no	maximal, Hall
25	C₂×D₅×Q₁₆	C₂×C₃	1	4	yes	—
26	D₅×C₄⋊Q₈	C₂×C₃	1	2	yes	—
27	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
28	#65470	C₂×C₃	1	4	yes	complemented
29	#65381	C₂×C₃	1	2	yes	complemented
30	#64988	C₂×C₃	1	1	yes	characteristic, complemented
31	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
33	C₃×Q₈×D₅	—	2	16	no	—
34	C₂×C₃×C₄×D₅	—	2	4	no	—
35	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
36	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×Dic₁₀	—	2	4	no	—
38	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
39	C₃×C₅×Q₁₆	—	2	8	no	—
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₃×C₅×C₄⋊C₄	—	2	4	no	—
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₃×#689	—	2	16	no	—
44	C₃×C₄⋊Dic₅	—	2	4	no	—
45	C₃×#683	—	2	8	no	—
46	C₃×C₄×Dic₅	D₄	1	2	yes	—
47	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₃×#168	C₂³	1	2	yes	—
49	C₃×Dic₂₀	—	2	8	no	—
50	C₃×C₈×D₅	D₄	1	8	yes	—
51	C₂²×C₃×Q₁₆	—	5	4	no	—
52	C₂×C₃×C₄⋊Q₈	—	5	2	no	—
53	C₃×#494	—	5	4	no	—
54	C₃×#494	—	5	2	no	—
55	C₃×#494	—	5	1	no	—
56	C₃×#494	D₅	1	1	yes	characteristic, direct product factor
57	C₂×C₃×C₄×C₈	—	5	1	no	—
58	C₂×Q₈×D₅	C₂²×C₃	1	4	yes	—
59	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	—
60	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
61	#4090	C₂²×C₃	1	4	yes	—
62	C₂×#689	C₂²×C₃	1	8	yes	—
63	D₅×Q₁₆	—	2	16	no	—
64	C₂×Dic₂₀	C₂²×C₃	1	4	yes	—
65	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
66	D₅×C₄⋊C₄	—	2	4	no	—
67	#4034	—	2	4	no	—
68	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
69	#4015	C₂²×C₃	1	2	yes	—
70	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₈×Dic₅	C₂²×C₃	1	2	yes	—
72	C₄×#168	C₂²×C₃	1	1	yes	characteristic
73	C₂×#494	—	5	1	no	Sylow
74	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
75	C₃×C₅×Q₈	—	2	8	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₂×C₃×Dic₅	—	2	4	no	—
80	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
82	C₃×C₄×D₅	—	4	8	no	—
83	C₃×C₄×D₅	Q₁₆	1	8	yes	complemented
84	C₃×C₄×D₅	C₂×D₄	1	4	yes	—
85	C₃×Dic₁₀	—	2	8	no	—
86	C₃×Dic₁₀	—	4	8	no	—
87	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
88	C₃×#168	C₂×D₄	1	4	yes	—
89	C₂²×C₃×Q₈	—	5	4	no	—
90	C₂×C₃×Q₁₆	—	5	8	no	—
91	C₂×C₃×Q₁₆	—	5	4	no	—
92	C₂×C₃×Q₁₆	—	10	16	no	—
93	C₂×C₃×Q₁₆	C₂×D₅	1	4	yes	—
94	C₂²×C₃×C₈	—	5	2	no	—
95	C₃×C₄⋊Q₈	—	5	4	no	—
96	C₃×C₄⋊Q₈	—	5	2	no	—
97	C₃×C₄⋊Q₈	—	10	4	no	—
98	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
99	C₂×C₃×C₄⋊C₄	—	10	4	no	—
100	C₂×C₃×C₄²	—	5	1	no	—
101	C₃×C₄×C₈	—	5	2	no	—
102	C₃×C₄×C₈	—	5	1	no	—
103	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
104	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
105	Q₈×D₅	—	2	16	no	—
106	C₂×C₄×D₅	—	2	4	no	—
107	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
108	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
109	C₂×Dic₁₀	—	2	4	no	—
110	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
111	C₅×Q₁₆	—	2	8	no	—
112	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
113	C₅×C₄⋊C₄	—	2	4	no	—
114	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
115	#689	—	2	16	no	—
116	C₄⋊Dic₅	—	2	4	no	—
117	#683	—	2	8	no	—
118	C₄×Dic₅	C₃×D₄	1	2	yes	—
119	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
120	C₂×#168	C₂³×C₃	1	2	yes	—
121	Dic₂₀	—	2	8	no	—
122	C₈×D₅	C₃×D₄	1	8	yes	—
123	C₂²×Q₁₆	—	5	4	no	—
124	C₂×C₄⋊Q₈	—	5	2	no	—
125	#494	—	5	4	no	—
126	#494	—	5	2	no	—
127	#494	—	5	1	no	—
128	#494	C₃×D₅	1	1	yes	characteristic, direct product factor
129	C₂×C₄×C₈	—	5	1	no	—
130	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₃×D₅	#127	1	2	yes	—
132	C₂×C₃×D₅	C₂×Q₁₆	1	4	yes	—
133	C₃×C₄×C₅	—	4	4	no	—
134	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	complemented
135	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
136	C₃×Dic₅	—	4	4	no	—
137	C₃×Dic₅	#127	1	2	yes	—
138	C₃×Dic₅	C₂×Q₁₆	1	4	yes	complemented
139	C₂×C₃×Q₈	—	5	4	no	—
140	C₂×C₃×Q₈	—	10	4	no	—
141	C₂×C₃×Q₈	—	10	16	no	—
142	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
143	C₂²×C₃×C₄	—	5	2	no	—
144	C₂²×C₃×C₄	—	5	1	no	—
145	C₂²×C₃×C₄	—	10	4	no	—
146	C₃×Q₁₆	—	2	8	no	—
147	C₃×Q₁₆	—	10	16	no	—
148	C₃×Q₁₆	—	10	8	no	—
149	C₂×C₃×C₈	—	5	2	no	—
150	C₂×C₃×C₈	—	5	8	no	—
151	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
152	C₃×C₄⋊C₄	—	2	4	no	—
153	C₃×C₄⋊C₄	—	10	8	no	—
154	C₃×C₄⋊C₄	—	10	4	no	—
155	C₃×C₄²	—	5	2	no	—
156	C₃×C₄²	—	5	1	no	—
157	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
158	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
159	C₅×Q₈	—	2	8	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
162	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
163	C₂×Dic₅	—	2	4	no	—
164	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
166	C₄×D₅	—	4	8	no	—
167	C₄×D₅	C₃×Q₁₆	1	8	yes	complemented
168	C₄×D₅	C₂×C₃×D₄	1	4	yes	—
169	Dic₁₀	—	2	8	no	—
170	Dic₁₀	—	4	8	no	—
171	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
172	#168	C₂×C₃×D₄	1	4	yes	—
173	C₂²×Q₈	—	5	4	no	—
174	C₂×Q₁₆	—	5	8	no	—
175	C₂×Q₁₆	—	5	4	no	—
176	C₂×Q₁₆	—	10	16	no	—
177	C₂×Q₁₆	C₂×C₃×D₅	1	4	yes	—
178	C₂²×C₈	—	5	2	no	—
179	C₄⋊Q₈	—	5	4	no	—
180	C₄⋊Q₈	—	5	2	no	—
181	C₄⋊Q₈	—	10	4	no	—
182	C₄⋊Q₈	C₂×C₃×D₅	1	2	yes	—
183	C₂×C₄⋊C₄	—	10	4	no	—
184	C₂×C₄²	—	5	1	no	—
185	C₄×C₈	—	5	2	no	—
186	C₄×C₈	—	5	1	no	—
187	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
188	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
189	C₂×C₃×C₅	C₂²×Q₁₆	1	2	yes	—
190	C₃×D₅	#494	1	4	yes	direct product factor
191	C₂³×C₃	—	5	1	no	—
192	C₃×Q₈	—	2	8	no	—
193	C₃×Q₈	—	10	8	no	—
194	C₃×Q₈	—	20	8	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	5	2	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	4	no	—
199	C₂×C₃×C₄	—	5	8	no	—
200	C₂×C₃×C₄	—	10	4	no	—
201	C₂×C₃×C₄	—	20	8	no	—
202	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
203	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
204	C₃×C₈	—	5	4	no	—
205	C₃×C₈	D₄×D₅	1	4	yes	—
206	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
207	C₂×D₅	C₃×#127	1	2	yes	—
208	C₂×D₅	C₂×C₃×Q₁₆	1	4	yes	—
209	C₄×C₅	—	4	4	no	—
210	C₄×C₅	C₂×C₃×Q₁₆	1	4	yes	complemented
211	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
212	Dic₅	—	4	4	no	—
213	Dic₅	C₃×#127	1	2	yes	—
214	Dic₅	C₂×C₃×Q₁₆	1	4	yes	complemented
215	C₂×Q₈	—	5	4	no	—
216	C₂×Q₈	—	10	4	no	—
217	C₂×Q₈	—	10	16	no	—
218	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
219	C₂²×C₄	—	5	2	no	—
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	10	4	no	—
222	Q₁₆	—	2	8	no	—
223	Q₁₆	—	10	16	no	—
224	Q₁₆	—	10	8	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	8	no	—
227	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	10	8	no	—
230	C₄⋊C₄	—	10	4	no	—
231	C₄²	—	5	2	no	—
232	C₄²	—	5	1	no	—
233	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
234	C₃×C₅	C₂×#494	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	5	2	no	—
236	C₂²×C₃	—	5	4	no	—
237	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
238	C₃×C₄	—	4	4	no	—
239	C₃×C₄	—	5	2	no	—
240	C₃×C₄	—	5	4	no	—
241	C₃×C₄	—	20	4	no	—
242	C₃×C₄	D₅×Q₁₆	1	4	yes	complemented
243	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
244	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
245	C₂×C₅	C₂²×C₃×Q₁₆	1	2	yes	—
246	D₅	C₃×#494	1	4	yes	direct product factor
247	C₂³	—	5	1	no	—
248	Q₈	—	2	8	no	—
249	Q₈	—	10	8	no	—
250	Q₈	—	20	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	5	2	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	5	4	no	—
255	C₂×C₄	—	5	8	no	—
256	C₂×C₄	—	10	4	no	—
257	C₂×C₄	—	20	8	no	—
258	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
259	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
260	C₈	—	5	4	no	—
261	C₈	C₃×D₄×D₅	1	4	yes	—
262	C₂×C₃	—	5	4	no	—
263	C₂×C₃	D₅×#127	1	1	yes	characteristic
264	C₂×C₃	C₂×D₅×Q₁₆	1	2	yes	—
265	C₅	C₂×C₃×#494	1	1	yes	characteristic, complemented, Sylow
266	C₂²	—	5	2	no	—
267	C₂²	—	5	4	no	—
268	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
269	C₄	—	4	4	no	—
270	C₄	—	5	2	no	—
271	C₄	—	5	4	no	—
272	C₄	—	20	4	no	—
273	C₄	C₃×D₅×Q₁₆	1	4	yes	complemented
274	C₄	C₂×C₃×D₄×D₅	1	2	yes	—
275	C₃	D₅×#494	1	1	yes	characteristic, direct product factor, Sylow
276	C₂	—	5	4	no	—
277	C₂	C₃×D₅×#127	1	1	yes	characteristic
278	C₂	C₂×C₃×D₅×Q₁₆	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#494	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
hypercenter	C₃×#494	D₅
hypocenter	C₅	C₂×C₃×#494
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#494

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃×C₅×#494	C₂	semidirect
4. C₃×#65470	C₂	semidirect
5. C₃×#65381	C₂	semidirect
6. C₃×#64988	C₂	semidirect
8. D₅×#494	C₃	direct (N × H)
27. C₅×#494	C₂×C₃	semidirect
28. #65470	C₂×C₃	semidirect
29. #65381	C₂×C₃	semidirect
30. #64988	C₂×C₃	semidirect
56. C₃×#494	D₅	direct (N × H)
83. C₃×C₄×D₅	Q₁₆	semidirect
128. #494	C₃×D₅	direct (N × H)
134. C₃×C₄×C₅	C₂×Q₁₆	semidirect
138. C₃×Dic₅	C₂×Q₁₆	semidirect
167. C₄×D₅	C₃×Q₁₆	semidirect
190. C₃×D₅	#494	direct (N × H)
210. C₄×C₅	C₂×C₃×Q₁₆	semidirect
214. Dic₅	C₂×C₃×Q₁₆	semidirect
234. C₃×C₅	C₂×#494	semidirect
242. C₃×C₄	D₅×Q₁₆	semidirect
246. D₅	C₃×#494	direct (N × H)
265. C₅	C₂×C₃×#494	semidirect
273. C₄	C₃×D₅×Q₁₆	semidirect
275. C₃	D₅×#494	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×D₅×Q₁₆	2
C₃×D₅×C₄⋊Q₈	2
C₃×C₅×#494	2
C₃×#65470	2
C₃×#65381	2
C₃×#64988	2
C₃×C₄×C₈×D₅	2
D₅×#494	3
C₂×C₃×#494	5

Subgroups — C3×D5×#494 #14438794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅×#494 #14438794